Discs are fictional objects that, when put on TwinBuilder's Phonograph, play the events of the Destroy the Godmodder series. There are three discs - Disc 1, Disc 2, and Disc 0. Disc 1 chronicles all of Destroy the Godmodder and Destroy the Godmodder 2 up to Scratch's Manor. Disc 2 chronicles Intermission 2 to midway through Trial 4. The remainder of Destroy the Godmodder 2 is not told on a disc, which is reflected by an increase in game corruption and the arrival of the Conflict. Disc 0 chronicles the entirety of Destroy the Godmodder 0rigins - and presumably all canon spinoffs.
TwinBuilder found Discs 1 and 2 in a record sleeve upon creating DTG2. Disc 0 was given to him at the end of the game by The Operator.
Whenever a disc is broken, it is usually caused by uncontrollable circumstances, such as a change in the Minecraft Forums' editor that makes it very hard to continue running the current game. This is represented in-universe by corruption and glitches.